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Trapped Gravitational Waves in Jackiw–Teitelboim Gravity

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Trapped Gravitational Waves in Jackiw–Teitelboim Gravity universe Article Trapped Gravitational Waves in Jackiw–Teitelboim Gravity Jeong-Myeong Bae 1, Ido Ben-Dayan 2,* , Marcelo Schiffer 2 , Gibum Yun 1 and Heeseung Zoe 1 1 School of Undergraduate Studies, College of Transdisciplinary Studies, DGIST, Daegu 42988, Korea; [email protected] (J.-M.B.); [email protected] (G.Y.); [email protected] (H.Z.) 2 Physics Department, Ariel University, Ariel 40700, Israel; [email protected] * Correspondence: [email protected] Abstract: We discuss the possibility that gravitational fluctuations (“gravitational-waves”) are trapped in space by gravitational interactions in two dimensional Jackiw–Teitelboim gravity. In the standard geon (gravitational electromagnetic entity) approach, the effective energy is entirely deposited in a thin layer, the active region, that achieves spatial self-confinement and raises doubts about the geon’s stability. In this paper we relinquish the “active region” approach and obtain self-confinement of “gravitational waves” that are trapped by the vacuum geometry and can be stable against the backreaction due to metric fluctuations. Keywords: gravitational waves; geons; Jackiw-Teitelboim gravity 1. Introduction In 1916, Einstein predicted that gravitational sources could produce waves of space- time from his theory of general relativity [1]. In 1955, Wheeler introduced a particle-like object, geon (gravitational electromagnetic entity), where gravitational perturbations are confined in space because of electromagnetic interaction [2]. He hoped to construct the geon Citation: Bae, J.-M.; Ben-Dayan, I.; as an elementary particle but that did not seem fruitful. Brill and Hartle elaborated this idea Schiffer, M.; Yun, G.; Zoe, H. Trapped by considering gravitational waves (GW) trapped by gravitational interactions [3], i.e., that Gravitational Waves in Jackiw– GW are somewhat localized in space by their self-interaction. Given the dispersive nature Teitelboim Gravity. Universe 2021, 7, of radiation, it seems such objects are metastable at best. Analyses in general relativity have 40. https://doi.org/10.3390/ devoted much effort to the discussion of whether such a solution is self-consistent and universe7020040 metastable [4–8]. These analyses assumed an empty asymptotic Minkowski background. Needless to stress the importance of considering stable self-confining gravitational config- Academic Editor: Claudia De Rham urations having as the background geometry the Friedman-Lemaitre–Robertson–Walker Received: 17 January 2021 (FLRW) Universe or at least, asymptotically de Sitter (dS). Significant works have been Accepted: 1 February 2021 done on asymptotic Anti-de Sitter in [9] and references therein. Published: 7 February 2021 In this paper, we study fluctuations of the gravitational field (“gravitational waves”) trapped in space by the vacuum geometry in the framework of classical two-dimensional Publisher’s Note: MDPI stays neu- (2D) Jackiw–Teitelboim (JT) gravity [10,11]. We prefer to use the term “trapped gravitational tral with regard to jurisdictional clai- ms in published maps and institutio- waves” instead of “geon” because in the classical geon solution the effective energy- nal affiliations. momentum that corrects the unperturbed solution is entirely deposited in a thin shell enclosing the geon (active region). Our motivation is to point towards a different kind of self-gravitating clump, a different paradigm that circumvents the need for an active region. Clearly, the choice of 2D gravity stems from the tremendous simplification of calculations. Copyright: © 2021 by the authors. Li- However, in 2D, the Einstein tensor vanishes identically and Einstein’s equations are trivial. censee MDPI, Basel, Switzerland. We therefore choose JT gravity as an alternative for a simple gravity theory in 2D that has a This article is an open access article cosmological constant (CC) and dynamical solutions. distributed under the terms and con- Generally speaking, in the vacuum, there is a competition between the gravitational ditions of the Creative Commons At- perturbations that travel at the speed of light and disperse, and their self-gravitational tribution (CC BY) license (https:// pull. The motivation of a non-vanishing CC comes from the intuitive fact that it further creativecommons.org/licenses/by/ generates a potential such that our solution sits in the vicinity of the potential minimum. 4.0/). Universe 2021, 7, 40. https://doi.org/10.3390/universe7020040 https://www.mdpi.com/journal/universe Universe 2021, 7, 40 2 of 18 This result corresponds to trapped “gravitational waves”. To fully understand the structure of the theory, we thoroughly discuss different gauges and independent degrees of freedom in the theory. More specifically, we study perturbations in the traceful gauge that is volume changing, and perturbations in traceless gauges, that better mimic GW gauges. Our analysis yields that perturbations can be trapped in some region of space. Furthermore, we discuss possible gauge issues, the connection between solutions in various coordinate systems and provide several examples. The paper is organized as follows. In Section2, we apply the method of [ 7] for finding gravitational geons to JT gravity. In Section3 we discuss possible gauge transformations and what degrees of freedom remain after using up the gauge freedom. In Section4, and Section5 we find analytic and numerical trapped solutions in various gauges. In Section6, we display the exact solution in the synchronous, conformal and spatially flat frame of references that exhibit a wave behavior and sketch similar trapped solutions. In Section7, we summarize our results and discuss future directions. 2. Finding a Geon in Jackiw-Teitelboim Gravity Our starting point is 2D gravity introduced by Jackiw and Teitelboim [10,11], where the equation of motion is given by R − L = 8pGT, (1) where R is the curvature scalar, L is the CC, T is the energy-momentum and c = 1. Notice that in 2D Newton’s constant is dimensionless and can always be absorbed into the gravitational field. As in [12,13], we take the following metric ansatz: gab = gab + hab, (2) where gab is the unperturbed metric with signature (−, +) and hab represents the perturba- tions. If we consider no matter, i.e., T = 0, Equation (1) becomes R(gab, hab) = L (3) Following [3,7], we expand it perturbatively as (0) (1) (2) R (gab) + R (gab, hab) + R (gab, hab) ' L (4) where (0), (1), (2), ... imply the orders in jhabj 1. We then solve this equation in three steps: First, the background geometry for the vacuum state comes from (0) R (gab) = L . (5) Second, the first order perturbation equation in hab (1) R (gab, hab) = 0 , (6) is a wave-type equation. Hence, the gravitational waves hab trapped in space are deter- mined by (6). Third, we test the stability of the solution by considering the backreaction on the metric through (0) (2) R (g˜ab) + hR (g˜ab, hab)i = L (7) where the original metric gab changes into g˜ ab due to the backreaction and h· · · i means time average. Universe 2021, 7, 40 3 of 18 3. Extraction of Physical Degrees of Freedom When considering perturbations off some background metric, it is important to prop- erly count the correct number of physical degrees of freedom that should be gauge invariant. Considering the general perturbed metric gab = gab + hab, the wave Equation (6) can be written as: 1 R(1)(g , h ) = hab − h˜ − h˜R(0)(g ) = 0, (8) ab ab ;ab 2 ab for any choice of coordinate system or gauge, where a semicolon denotes a covariant deriva- ˜ ab tive, is the covariant D’Alambertian, and h ≡ g hab is the trace. This expression seems to suggest that for the purpose of calculations there are two preferred gauges, traceless and Lorentz. This is a proper time for pausing the calculations and discussing the gauge freedom in 2D. Coordinate transformations can be represented by gauge transformations 0 hab = hab − xa;b − xb;a (9) a for any vector x , being of the same order of magnitude of hab itself. As a side remark, solving the field equations for 0 1 h = h − g h˜ (10) ab ab 2 ab as it is usually done in 4D, is pointless as this relation cannot be inverted to obtain hab since a in 2D the trace h ≡ ha vanishes identically. This is somewhat reminiscent of the fact that the Einstein tensor is trivial in 2D. Consequently, all the discussion of gauge invariance must be done in terms of hab. In 2D we can always express any vector as ,b xa = f,a + eaby (11) for two different scalar fields f, y and eab stands for the Levi–Civitta in 2D q eab = (−g)[a, b], (12) where [0, 1] ≡ 1; [0, 0] = [1, 1] ≡ 0; [1, 0] ≡ −1 (13) 3.1. Lorentz Gauge It is always possible to make a gauge transformation which brings a general perturba- tion hab to the Lorentz gauge. Consider the divergence of a desired gauge transformation, 0 b = b − ;b − b = h a;b ha ;b xa ;b x; ab 0. (14) The commutation of covariant derivatives satisfies b b d x;ba − x;ab = −Radx . (15) Furthermore, recall that in 2D 1 R = Rg . (16) ab 2 ab In order to bring a generic perturbation to the Lorentz gauge we have to solve: 1 x ;b + xb + Rx = h b , (17) a ;b ;ba 2 a a ;b Universe 2021, 7, 40 4 of 18 and in terms of the aforementioned scalar fields, ,d b b xa = f,a + eady ) x ;b = f ) x ;ba = (f),a (18) ;b = bc = bc( + d ) = bc( + d ) xa ;b g xa;bc g f;abc ea y;dbc g f;bac ea y;bdc (19) Recalling that for any vector la d lb;ac − lb;ca = Rbacld, (20) then ;b = ( ) + Rd + d( ) + dRl xa ;b f ,a a f,d ea y ,d ea dy,l. (21) In view of (6) and the fact that the curvature is constant, R d R 2f + f + ea (y) + y = va, (22) 2 ,a 2 ,d = ;b = + R = + R where va ha ;b.
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